Expressions such as 10^{12} or 2^{50} may not denote natural numbers under strict finitist criteria

On the other hand, such indirect reference to polynomial rates of growth is avoided in similar functional characterizations of \(\textbf{FP}\) due to Leivant (1994) (using a form of positive second-order definability over strings) and Bellantoni and Cook (1992) (using a structural modification of the traditional primitive recursion scheme). Direct reference to polynomial rates of growth is also avoided in the formulation of the first-order arithmetical theory now known as \(\text{I}\Delta_0\) (w